Slow light with high normalized delay-bandwidth product in low-dispersion photonic-crystal coupled-cavity waveguide

Accepted Manuscript Slow light with high normalized delay-bandwidth product in low-dispersion photonic-crystal coupled-cavity waveguide Israa Abood, Sayed Elshahat, Karim Khan, Luigi Bibbò, Ashish Yadav, Zhengbiao Ouyang

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S0030-4018(19)30071-9 https://doi.org/10.1016/j.optcom.2019.01.063 OPTICS 23824

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Optics Communications

Received date : 16 October 2018 Revised date : 8 January 2019 Accepted date : 24 January 2019 Please cite this article as: I. Abood, S. Elshahat, K. Khan et al., Slow light with high normalized delay-bandwidth product in low-dispersion photonic-crystal coupled-cavity waveguide, Optics Communications (2019), https://doi.org/10.1016/j.optcom.2019.01.063 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Slow light with high normalized delay-bandwidth product in lowdispersion photonic-crystal coupled-cavity waveguide Israa Abood, Sayed Elshahat, Karim Khan, Luigi Bibbò, Ashish Yadav and Zhengbiao Ouyang* College of Electronic Science and Technology of Shenzhen University, THz Technical Research Center of Shenzhen University, Key Laboratory of Optoelectronics Devices and Systems of Ministry of Education and Guangdong Province, Shenzhen University, Shenzhen 518060, China *Corresponding author: [email protected]

Abstract Slow-light performance based on photonic-crystal coupled-cavity waveguide (PC-CCW) is investigated with low-dispersion pulse transmission. A chain of periodically arranged specific defect cavities of air holes are created along the axis of the waveguide and each of the cavities looks like an irregular hexagon with arrow-like left and right sides to form the PC-CCW. By changing the radius and position of certain defects, slow light modes are achieved. Through optimization, we obtained a highest compromise value for an operating bandwidth of 12.9 nm with an average group index of 70.75 and a normalized delay-bandwidth product (NDBP) of 0.5873. This value of NDBP with low group velocity dispersion is higher than that in other structures based on waveguides or coupled cavities reported previously. The arrow-like shape of the cavity sides is shown to be perfect for reducing the transmission loss of the waveguide. The proposed structure with high NDBP is beneficial for buffering applications and slow light devices. Keywords Slow light; Coupled cavity waveguide; Dispersion 1. Introduction Photonic crystals (PhCs) are predominantly artificial materials with periodic distribution of dielectric constant [1]. They have promising characteristics, in particular, for slow light [2, 3] that can off er the possibility for compressing optical pulses, which reduces device footprint and enhances light–matter interactions [4, 5]. Though a light with a frequency in the PhC band gap (PBG) cannot propagate inside the PhC, the periodicity of the PhC dielectric structure (and, correspondingly, band gap) can be modified by introducing some defects to split the PBG, then the light with a frequency in the PBG split can be confined in the defect area and also the light can propagate through PhC with low group velocity and extinction losses through resonant tunneling [6]. Resultantly, many slow-light PhC structures have been extensively utilized in different applications, for instance, optical delay lines [7, 8], all-optical buffering [9, 10], optical storages [11] and optical switches [12] . Nevertheless, slow light effect with low group velocity, high group index, accompanies with group velocity dispersion (GVD) [12], that causes pulse broadening, high losses, and narrow transmission bandwidth [13]. Therefore, it is a great challenge to propose slow-light PhC structures with high group index, negligible GVD, and lossless transmission bandwidth. In comparison with linedefect photonic-crystal waveguide (W1 PCW), the photonic-crystal coupled-cavity waveguide (PCCCW) has an ample feature for slow light property [14, 15]. Various PhC structures are reported for improving the slow light performance by adjusting structural parameters for instance, changing the air hole radii [16] and their positions [17], using air rings for the air holes [18], shifting the first and second rows of air holes symmetrically or asymmetrically [19], inserting successive defect rods in an intrinsic PCW [20], and introducing three line defects in PCW [21]. However, further study is required for achieving desired application performance. In this paper, we propose a structure based on PCCCW by inserting specific defects of air holes periodically set along the axis of the waveguide, the defects form cavities, and each cavity looks like an irregular hexagon with arrow-like left and right

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sides. By changing the radius and position of certain defects, slow light modes are achieved with a high value of normalized delay bandwidth product (NDBP) about 0.5873, which is up to date the highest value of NDBP as compared with that reported in literature above mentioned. The NDBP tells a compromise between the available operating bandwidth and group index, and is an adequate indicator for slow light devices and high performance buffering applications. 2. General formulae for calculating slow light properties In order to analyze and enhance slow wave properties, some definitions and formulae need to be clarified. The slow light properties can be evaluated by group index n g ,which is the ratio of the light speed in vacuum C to the group velocity vg , so [22]

ng 

d neff dK C  neff    vg d dU

(1)

where U  a / 2 C  a /  and K  ka / 2 are the normalized frequency and normalized wave vector, respectively with lattice constant a and operating wavelength  . The group velocity vg of the guided mode is defined as the derived number of angular frequency  with respect to wave vector k , i.e., vg  d  / dk , and the effective refractive index is defined as neff  k  / 2 . Moreover, GVD as a discriminatory for medium dispersion and is defined as: GVD 

a dng . 2 C 2 dU

(2)

As indicated by Eq. (2), the dependence of group velocity or group index on frequency is identified by the GVD parameter that leads to pulse broadening and distortion [23, 24]. Due to the group velocity (index) dispersion, the transmitted pulse that carries the bit data will become distorted along the propagation path. Therefore, for less distortion, we need to decrease the GVD in transmission. On the other hand, a proper GVD can be used to compensate pulse distortion. For compensation purposes, we need to design the compensating-device GVD and the input wave GVD to have opposite signs. Moreover, a slow-light is generally transmitted in a waveguide with intrinsic dispersions. Therefore, designing a wide zero-dispersion band in a slow waveguide system is an important task for high fidelity transmission of light signals. Hence, it is necessary and crucial to study the GVD property for slow light transmission. Due to the inevitable trade-off between the slow light bandwidth and the obtained group index, NDBP is a more suitable parameter for balance consideration and is more often used when the devices have different lengths and/or different operating frequencies [3, 25]. NDBP is calculated as [26] Δ NDBP  ng  , (3) 0 where 0 is the central frequency over bandwidth Δ and n g is the average group index restrained over ±10% of n g variation [27]. The average group index is calculated by

ng 

1 N  ngi , N i

(4)

where N is the number of frequency points calculated in simulation at the operating frequency band and ngi is the group index for each frequency point calculated in the frequency band for the variation of n g within ±10%.

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3. Coupled cavity slow-light waveguide design and analysis The PC-CCW is created in a triangular-lattice PhC, consisting of circular air holes of radius R = 0.25a on a silicon slab of refractive index 3.48. To form the PC-CCW in the PhC, every two consecutive air holes are removed while keeping the third one along the center axis (forming coupled cavities arranged in such a way that the cavities are periodically distributed along the axis of the waveguide with a period of 3a). Each air hole in the waveguide center and the two neighboring air holes in the upper and lower first rows on the waveguide walls form a shape looking like an arrow, i.e., the right and left sides of the cavity have an arrow-like shape, as shown in Fig. 1. The defects in the proposed structure have two different radii ( r1 , r2 ) where r1 is the radius of arrow defect and r2 is the radius of the remaining air holes in the first rows on the upper and lower waveguide walls. The dashed black rectangle in Fig. 1 shows a specific supercell which can be considered for the plane wave expansion (PWE) method calculations [28] by using the BANDSOLVE module of software Rsoft [29] with a unit cell period of (Λ = 3a).

Fig. 1. Schematic structure of the proposed PC-CCW geometry with a triangular-lattice on a silicon slab of index n = 3.48 with air hole radius R = 0.25a. The dashed black rectangle displays a specific supercell that has been used for PWE calculations, where Λ = 3a. The air holes (in white) are shifted by x along the propagation direction.

Figure 2 shows the slow light properties (ng, GVD) with normalized frequency and dispersion curves for hexagonal lattice of air holes in dielectric slab likewise dielectric rods in air background at R  r1  r2  0.25a with x  0.00a . The dispersion curve of the slow light guided mode for hexagonal lattice of air holes in dielectric slab only supports TM polarized mode as shown by the inset in Fig. 2 (a). Under constant group index with zero dispersion for flat band norms from the curves of (ng, GVD), an average group index n g of 70.5 ranging from 0.21896 (2 C / a) to 0.2196 (2 C / a) is obtained, with a bandwidth of 4.514 nm and an NDBP of 0.2053. However, the hexagonal lattice of rods in air background supported both TE and TM polarized modes as shown by the inset in Fig. 2 (b). For TE mode, an average group index n g is obtained as 12.53 in the frequency range from 0.3358 (2 C / a) to 0.3427 (2 C / a) , with a bandwidth of 31.65 nm and an NDBP of 0.2558. For TM mode, an average group

index n g is obtained as 30.48 in the frequency range from 0.4089 (2 C / a) to 0.4116 (2 C / a) , with a bandwidth of 10.28 nm and an NDBP of 0.2022. Therefore, for operating in TM modes, it is better to use the air-hole-in-dielectric-slab structure, and for operating in TE modes, it is better to choose the dielectric-rod-in-air structure. One advantage for using the triangle-lattice PhC is that it can supply a

wider PBG and is less sensitive to geometrical-parameter variations than square-lattice PhCs [30], while a wider PBG is the basic condition for a larger transmission band of the PCW. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 2. Slow light properties (ng, GVD) versus normalized frequency for the case with hexagonal lattice of (a) air holes in dielectric slab and (b) dielectric rods in air background. The insets show the dispersion curves at R  r1  r2  0.25a with

x  0.00a

Characteristically, we will concern on erecting the slow-light performance on point and line defects simultaneously for the triangle-lattice PhC of air-hole-in-dielectric-slab, in order to obtain slow guided modes in the proposed PC-CCW with high group index and low GVD combined with high NDBP. In the following, by detecting the modes of slow light and through Eqs. (1) and (2) we extract the group indices and GVD by PWE method. Then, we analyze theoretically the data results for higher NDBP. We first change r1 in a certain range for highest NDBP and lowest-dispersion pulse transmission. In a further step we optimize r2 in a certain range while keeping r1 to be the value obtained in the previous step for the highest value of NDBP. Afterward, we optimize further the proposed structure by changing x in a certain range for the arrow sides while, keeping the values of r1 and r2 at optimized statues obtained in the previous step. Finally, we use the finite-difference time domain (FDTD) method in the optimized structure to confirm the low-distortion pulse transmission.

Fig. 3. Slow light properties affected by the radius r1 in the range of 0.15a  r1  0.22a with R and r2 fixed at 0.25a and x at 0.00a; (a) Dispersion curves of guided modes; (b) group index with normalized frequency; (c) GVD with normalized frequency.

Figure 3 declares the slow light properties affected by the radius r1 in the range of 0.15a  r1  0.22a with an increment of 0.01a with R  r2  0.25a and x = 0.00a. The dispersion curves are obtained for transverse magnetic (TM) polarized mode with nonzero (Hy, Ex, Ez). Figure 3(a) shows the dispersion curves of guided modes for different values of r1 . We can see obviously that the calculated dispersion curves

move up to the higher-frequency area with increasing r1 . This is understandable as large r1 means 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

small value of equivalent refractive index of the structure and the response frequency of the structure increases according to the scaling properties of Maxwell equations. The ng and GVD can be picked up from Eqs. (1) and (2) respectively. Accordingly, the flatter bandwidth is obtained for low slope of dispersion curve and also the dispersion curve is linear in a wide frequency range. The variation of ng and GVD with normalized frequency are shown in Figs. 3 (b) and 3(c). It can be seen from Figs. 3 (b) and 3(c) that the flat band approaches maximum at r1  0.18a for

R  r2  0.25a with an average group index ng about 31.32 for ±10% variation of ng combined with a wider bandwidth of 17.2 nm (bandwidth  at 1550 nm). The group index is inversely proportional to the bandwidth for a given structure [31]; therefore, NDBP is indispensable for quantifying the slow light performance as shown in Eq. (3), and it is defined as the product of average group index ng and the normalized bandwidth   / 0  . Consequently, higher compatible values of group index and normalized bandwidth are achieved for NDBP about 0.3474 at r1  0.18a . It’s the optimum value in this step and is used for further optimization of other parameters. An additional factor for describing slow light property is GVD that causes pulse distortion in the propagation process. As shown in Fig. 3(c), GVD values vary from 3 106 to 3 106 . This shows the possibility of dispersion compensation [32] by using our proposed structure. Therefore, in the following we can fix the value of r1 at the optimum value about 0.18a. For further optimization, r2 changes in the range 0.22a  r2  0.29a with an increment of 0.01a, while keeping the value of r1 at 0.18a and x = 0.00a. The defect air holes form cavities according to the photonic crystal cavity theory

[33], and the defect cavities combine together to form the line defect cavities. In this case, the slow-light performance is studied on point and line defects simultaneously. Slow light properties of ng and GVD are shown in Figs. 4(a) and 4(b), respectively. From Fig. 4(a) it can be seen that the group index value increases rapidly with increasing r2 , simultaneously the flat band of zero dispersion decreases sharply as shown in Fig. 4(b). We can explain these results as follows. Each hole in the first row with radius r2 can be regarded as a defect cavity (sub cavity) besides the irregular hexagonal defect cavity consisting of four holes with radius r2 and four holes with radius r1 . As larger r2 corresponds to less value of effective refractive index in the cavity area, we can expect that the resonance frequencies (corresponding to the lowest group velocity in the group velocity spectra) will shift to the high frequency region as r2 increases, and at the same time the group velocity will become higher. To understand the bandwidth changing, it is necessary to consider the coupling among the small sub cavities and the large irregular hexagonal cavities. When r2 increases, the resonance frequencies of the small sub cavities and the large irregular hexagonal cavity will have larger difference. As a result, the band that can allow the wave to pass the whole structure will become smaller. When r2  0.27a with the average group index ng of 66.33, it has a bandwidth about 9.51 nm with the highest NDBP of 0.407.

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Fig. 4. Slow light properties of guided modes affected by r2 in the range of 0.22a  r2  0.29a with R, r1 and x fixed at 0.25a, 0.18a and 0.00a respectively; (a) group indexes with normalized frequency; (b) GVD with normalized frequency.

Fig. 5. Slow light properties of guided modes for x changing in the range 0.40a  x  0.50a with r1 , r2 fixed at 0.18a, 0.27 a , respectively (a) Group index with normalized frequency; (b) GVD with normalized frequency.

Table 1. Slow-light properties of guided modes affected by x in the range 0.40a  x  0.50a with ( r1 , r2 ) fixed at (

0.18a, 0.27 a ). x /a

 at 1550 nm

ng

NDBP

0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5

19.3 17.8 16.3 15.1 14.5 12.9 11.3 9.17 8 7.43 7.16

35.92 38.57 42.71 48.9 59.21 70.75 75 78.31 76.07 77.39 80.57

0.4469 0.4418 0.4481 0.4759 0.5531 0.5873 0.5475 0.4634 0.3926 0.3709 0.3724

Finally, for accomplishment of the optimization, we keep r1 and r2 as the optimum values of 0.18a and 0.27a respectively and seek the optimum value of x in the range of 0.40a  x  0.50a with 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

an increment of 0.01a. The influence of x on ng and GVD is shown in Fig. 5 and Table 1. For application in the communication waveband, Table 1 shows the results from Fig. 4 when the lattice constant a is varied to be the value that the central wavelength is fixed at 1550nm. As can be seen clearly from Fig. 5 and Table 1, the bandwidth centered at 1550 nm decreases from 19.3 nm to 7.16 nm as the corresponding average group index increases from 35.92 to 80.57. From Fig. 5(a) it can be seen that the group index increases gradually and the shape of curve is usually U-shape for 0.40a  x  0.43a and 0.46a  x  0.50a , however for 0.44a  x  0.46a the shape of the curve becomes unusual. Most of the curves look like a combination of two transmission bands, similar to that described in the theory of transmission line [21]. The fact is that the coupling of the sub cavities and the irregular hexagonal cavities changes as x varies. In addition, Fig. 5(b) indicates that the GVD fluctuating symmetrically from (-2 106 a / 2 C 2 ) (negative dispersion) to (+2 106 a / 2 C 2 ) (positive dispersion), which can be used for dispersion compensations [32, 34]. If the absolute value of GVD below (106 a / 2 C 2 ) is regarded as low GVD, we can consider our proposed PC-CCW as a low GVD device [35]. As usual, Table 1 shows a trade-off between the bandwidth and the group index, then we consider a balanced case for ±10% variation of ng and obtain x  0.45a as an optimum value. Table 1 shows that, at the optimum value of x  0.45a , the bandwidth is 12.9 nm combined with ng  70.75 and the highest value of NDBP is about 0.5873. The proposed PC-CCW provides a higher value of NDBP compared with other structures based on waveguides or coupled cavities reported previously as shown in Table 2. Table 2. Average group index, bandwidth  at 1550 nm and NDBP compared with previous works. Ref.

ng

(Current work) 70.75 [36] 40 [21] 11.7 [37] 20.1 [38] 44 [39] 54 [20] 28 [40] 647

 at 1550 nm NDBP 12.9 6.0 65.7 36.8 11.0 12.7 26.5 None

0.5873 0.16 0.496 0.469 0.312 0.442 0.48 0.54

4. Time domain analysis of slow light propagation To further demonstrate the results obtained by the PWE method, time-domain optical pulse transmission is investigated through the proposed PC-CCW with 2D FDTD method by using FULLWAVE module of software Rsoft [29]. The optimization is considered in FDTD simulation with average group index ng  66.33 for r1  0.18a , r2  0.27a and x=0.00a by the PWE method. In FDTD, normalized filed intensity can be observed by two monitors for Gaussian pulse through a propagation length about 50a for the proposed PC-CCW. The first one is the input monitor located at 5a from the Gaussian pulse source and the second one is the output monitor located at 45a from the input, so the transmission length is 40a.

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Fig. 6. Time-domain optical pulse transmission in the proposed PC-CCW for r1  0.18a , r2  0.27a , and x=0.00a ; (b) magneticfield profile transmission of light pulse inside the optimized PC-CCW.

The average group index is calculated by [41] ng  C T / L , where CT represents the normalized time delay between input and output pulses through propagation length L=40a. A Gaussian pulse source with a center wavelength of 1550 nm was launched for FDTD calculation with a = 319.5 nm. Figure 6(a) reveals the simulation result of the time-domain optical pulse transmission from the input to the output monitors. The peaks of the normalized field intensity for input and output are observed at 2238 CT and 3007 CT respectively, so the delay time between them is 769 CT. Consequently, the average group index is ng  60 . The minor disparity between the two values of group index obtained by PWE and FDTD is primarily originated from the limited discretization and area in FDTD simulations and the number of plane waves for calculation in the PWE method which deals with an infinite-area structure [42]. The normalized Full-Width at HalfMaximum (FWHM) of the input and output pulses monitored are 660 CT and 702 CT respectively, and the relative pulse distortion per period length is obtained as 0.15%/ a , which confirms that our proposed PCCCW has low distortion for optical pulse transmission. Therefore the pulses can be transmitted along the waveguide deprived of obvious distortion. The comparison shows that the results obtained from the two different methods agree with each other, so it proves that the numerical results are reliable. Figure 6(b) illustrates the propagation of the wave package inside the designed PC-CCW and the field profile is totally different from that in conventional PCWs. Periodic envelop of optical field distribution are observed with spatial light confinement in the cavities lengthways along the waveguide, propagating from one cavity to the other. On account of each defect can be regarded as a cavity according to the theory of photonic crystal cavities [33]. Introducing a series of defect air holes on the upper and lower walls of the PCW produces a series of cavity loads as energy reservoirs of electromagnetic (EM) energy in the system. Therefore, the wave can be localized in the cavities and more cavities mean less propagation speed of waves in the waveguide as more time has to be taken to fill energy to more cavities. To balance between the bandwidth and the time delay due to the energy store process, between two neighboring defect cavities are two regular holes which have the same radius as that of the background holes in the photonic crystal. The holes in the center of the waveguide is for coupling of waves among the cavities. For the arrow-like structure, looking from the front side it likes an emitting antenna with a convex shape being favor of radiating waves, while looking from the backside it likes a receiving antenna with a concave shape being favor of collecting waves. Therefore, successive emitting and collecting micro antennas are created in the system, leading to a high coupling efficiency for the wave transmission the structure. Figures 6(a) and 6(b) show clearly that the power loss in the propagation

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process is very small, demonstrating that the arrow-like shape of the cavity sides is perfect for reducing the transmission loss of the waveguide, while defects will generally cause large wave reflections and thus large transmission loss in conventional structures. Finally for fabrication, we point out that when the design is completed, the structure and all the holes are fixed, so that no shift is required in the fabrication process. For fabrication of 2D-PhCs [43, 44], the optimum diameters were fine-tuned by adjusting the exposure time of the electron beam lithography [17]. Generally, a mask is first generated according to the designed structure, then one can use the mask in the lithography process to obtain the designed structure precisely. 5. Conclusion A photonic-crystal coupled irregular-hexagon cavity waveguide is investigated with low-dispersion pulse transmission, including point defect and line defect simultaneously. By inserting an irregular hexagonal defect cavity through removing a certain number of air holes from the center of the PhC with shifting of air holes on the arrow-like sides of the irregular hexagonal cavity. Through optimization of parameters, we obtained a highest value of the operating bandwidth about 12.9 nm and its correlating average group index of 70.75. It allows for a high NDBP value of 0.5873 with low GVD. The arrow-like shape of the cavity sides is perfect for reducing the transmission loss of the waveguide. Furthermore, we confirmed the results obtained by PWE method through FDTD simulations of light pulse transmission in the structure. Acknowledgment This work is supported in part by the National Natural Science Foundation of China (NSFC) under Grants 61275043, 61307048, 60877034, and 61605128, in part by Guangdong Province Natural Science Founds (GDNSF) under Grant 2017A030310455, and in part by the Shenzhen Science Funds (SZSF) under Grants JCYJ20170302151033006 and 20180123. References [1] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett., 58 (1987) 2059-2062. [2] T.F. Krauss, “Why Do We Need Slow Light?”, Nat. Photonics, 2 (2008) 448–449 [3] T. Baba, Slow light in photonic crystals, Nat. Photonics, 2 (2008) 465. [4] T.F. Krauss, Slow light in photonic crystal waveguides, J. Phys. D: Appl. Phys., 40 (2007) 2666-2670. [5] Y.A. Vlasov, M. O'Boyle, H.F. Hamann, S.J. McNab, Active control of slow light on a chip with photonic crystal waveguides, Nature, 438 (2005) 65. [6] M. Notomi, Strong Light Confinement With Periodicity, Proceedings of the IEEE, 99 (2011) 1768-1779. [7] M.L. Povinelli, S.G. Johnson, J.D. Joannopoulos, Slow-light, band-edge waveguides for tunable time delays, Opt. Express, 13 (2005) 7145-7159. [8] M. Danaie, A. Geravand, S. Mohammadi, Photonic crystal double-coupled cavity waveguides and their application in design of slow-light delay lines, Photonics Nanostruct. Fundam. Appl., 28 (2018) 61-69. [9] R.S. Tucker, P.C. Ku, C.J. Chang-Hasnain, Slow-light optical buffers: Capabilities and fundamental limitations, J. Lightwave Technol., 23 (2005) 4046-4066. [10] C. Li, Y. Wan, W. Zong, High sensitivity electro-optic modulation of slow light in ellipse rods PC-CROW, Opt. Commun., 395 (2017) 188-194. [11] M. Merklein, B. Stiller, K. Vu, S.J. Madden, B.J. Eggleton, A chip-integrated coherent photonic-phononic memory, Nat. Commun., 8 (2017) 574. [12] K. Fasihi, High-Contrast All-Optical Controllable Switching and Routing in Nonlinear Photonic Crystals, J. Lightwave Technol., 32 (2014) 3126-3131. [13] R.W. Boyd, D.J. Gauthier, Slow and Fast Light, Progress in Optics, 43 (2001) 363-385. [14] A. Yariv, Y. Xu, R.K. Lee, A. Scherer, Coupled-resonator optical waveguide:a proposal and analysis, Opt. Lett., 24 (1999) 711-713. [15] K. Sakai, E. Miyai, S. Noda, Two-dimensional coupled wave theory for square-lattice photonic-crystal lasers with TM-polarization, Opt. Express, 15 (2007) 3981-3990.

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